cal analysis, and this disagreement can be traced back through
the centuries to the Middle Ages and thence to ancient Greece.
All sides have had their representatives in all ages of mathe-
matical thought, whether that thought was disguised in provo-
cative paradoxes, as with Zeno, or in logical subtleties, as with
some of the most exasperating logicians of the Middle Ages.
The root of these differences is commonly accepted by mathe-
maticians as being a matter of temperament: any attempt to
convert an analyst like Weierstrass to the scepticism of a
doubter like Kronecker is bound to be as futile as trying to
convert a Christian fundamentalist to rabid atheism,
A few dated quotations from leaders in the dispute may serve
as a stimulant - or sedative, according to taste — for our enthu-
siasm over the singular inteBectual career of Georg Cantor,
whose 'positive theory of the infinite* precipitated, in our own
generation, the fiercest frog-mouse battle (as Einstein once
called it) in history over the validity of traditional mathema-
In 1831 Gauss expressed his 'honor of the actual infinite' as
follows. 'I protest against the use of infinite magnitude as some-
thing completed, which is never permissible in mathematics.
Infinity is merely a way of speaking, the true meaning being a
limit which certain ratios approach indefinitely close, while
others are permitted to increase without restriction*'
Thus, if x denotes a real number, the fraction I/as diminishes
as x increases, and we can find a value of a? such that l/# differs
from zero by any preassigned amount (other than zero) which
may be as small as we please, and as x continues to increase, the
difference remains less than this preassigned amount; the limit
of I/a, 'as x tends to infinity,* is zero. The symbol of infinity is
oo; the assertion l/oo = 0 is nonsensical for two reasons: *divi-
sion by infinity' is an operation which is undefined) and hence
has no meaning; the second reason was stated by Gauss.
Similarly 1/0 = oo is meaningless.
Cantor agrees and disagrees with Gauss. Writing in 1886 OB
the problem of the actual (what Gauss called completed) infi-
nite, Cantor says that 'in spite of the essential difference
between the concepts of the potential and the actual 44rnfiniteT%